Mark Kac (1914-1984), in the introduction to his autobiography Enigmas of Chance, writes:

In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the "ordinary" and the "magicians". An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students because they cannot be emulated and it must be terribly frustrating for a brilliant young mind to cope with the mysterious ways in which the magician's mind works.

But is this true? Or do the "magicians", as Kac terms them, simply tap into different sources of power on some deep level? Do they perhaps exploit different ways of thinking --- highly nonverbal mental functions which they can't explain but which could, by study and emulation, be applied by others? And on the other hand, are the "ordinary" geniuses actually extraordinary in ways which tend to escape notice?

How easy is it to be "many times better" than we ordinarily are? Might there not be computationally complex processes going on, beneath the areas of thought open to introspection? Processes which grow exponentially hard, so that speeding up the mental processor's clock rate could never actually succeed in performing them within practical amounts of time?

What are the ultimate limits of "intelligence"? If one had arbitrarily huge memory and similarly huge computational speeds, would one's "effective IQ" still max out somewhere around 1000 or so? Are there essential mental processes, perhaps involving pattern-matching and function evaluation, that are like the classic NP tasks of computer science --- problems which brute force cannot efficiently solve?